validation of a nonlinear reactive control law for three ... · (ni 9401). the analog input module...

Validation of a nonlinear reactive control law for three-dimensionalparticle tracking in confocal microscopy

Trevor T. Ashley†, Catherine Chan-Tse?, and Sean B. Andersson†‡† Dept. of Mechanical Engineering, ?Dept. of Electrical and Computer Engineering, ‡ Division of Systems Engineering

Boston University, Boston, MA 02215 USA{tashley,cchantse,sanderss}@bu.edu

Abstract— A nonlinear control law for reactively trackinga diffusing fluorescent particle in three dimensions is reviewedand preliminary experimental results are presented. The controllaw, originally derived in the context of exploring scalarpotential fields, converges to a volume around the maximumof the point spread function of a confocal microscope usingvery little information of the underlying structure of thefield. Experimental results show the control law can trackparticles with diffusion coefficients over 3 µm2/s with moderateexcitation intensities.

I. INTRODUCTION

Over the past decade, single molecule spectroscopy andimaging has given researchers tremendous insight into vari-ous biological phenomena. Some notable results include thestudy of influenza [1], hepatitis B [2], and molecular motors[3]. Most single particle methods rely on wide-field imagingusing charge coupled device (CCD) cameras. Despite theirsuccesses to date, this reliance limits the achievable sensitiv-ity as well as the spatial and temporal resolution, especiallywhen studying single particles inside living cells.

To alleviate the inherent problems of wide-field detection,several alternative methods have been developed. One suchapproach split a laser into two beams, focused each beamonto separate focal planes, and rotated the beams about asmall circle [4]. When a particle was “locked” near the centerof the circle, the position of the particle could be inferredusing lock-in demodulation of the measured fluorescenceintensity. The position estimate was used to regulate theposition of a three-axis nanopositioning stage to keep theparticle close to the center of the circle. Other techniquesuse the same principle of servoing about a position estimatebut have different methods for localizing the particle. Forexample, earlier work by one of the authors (with others)took several intensity measurements in a constellation patternand used GPS-like localization to calculate the position ofthe particle [5]. Two other methods use multiple photondetectors to determine the position of the particle [6], [7].All of these techniques have demonstrated the ability oftracking diffusing particles with coefficients between 0.1 and20 µm2/s at various signal-to-noise ratios (SNRs). Thesetechniques, however, either directly or indirectly measure thedisplacement of the particle, thereby leading to costly optics,complicated configurations, or temporal inefficiencies.

In light of these concerns, a three-dimensional nonlin-ear reactive tracking algorithm was developed in [8] that

avoids real-time localization while forcing the trajectory ofthe excitation volume to converge to a volume around theparticle. The algorithm avoids temporal inefficiencies andis versatile in the fact that it can be implemented on anydevice that serially acquires measurements of any three-dimensional scalar potential field of arbitrary shape (e.g.magnetic force [9], multiphoton, and stimulated emissiondepletion microscopy). Since the particle position is not mea-sured in real-time, a variety of techniques can be employedto analyze the experimental data and determine particletrajectories and statistical information. In this paper, we usethe fluoroBancroft estimation algorithm introduced by oneof the authors [10] to estimate the particle position andthen analyze the resulting trajectories to determine diffusioncoefficients.

II. EXPERIMENTAL SETUP

Our experimental setup, illustrated in Fig. 1, consistedof a standard epi-fluorescence confocal microscope builtupon an inverted microscope (Axiovert 200, Carl Zeiss). A488 nm laser (FiberTec II, Blue Sky Research) was directedthrough a beam expander and onto a dichroic filter (T495LP,Chroma) which reflected the beam onto the back apertureof an objective lens (water immersion, 63x, 1.2 N.A. C-Apochromat, Carl Zeiss). The objective lens focused thebeam to a diffraction-limited spot onto the sample. A three-axis piezoelectric nanopositioning stage (Nano-PDQ, MadCity Labs), driven by a high-voltage amplifier (Nano-Drive85, Mad City Labs), carried the sample and provided 50 µmof fine displacement in each axis. The fluorescence emittedby the sample was collected by the objective lens and passedthrough the dichroic filter and a bandpass filter (HQ625/30m,Chroma). The output signal was then split into two beams bya beamsplitter which focused a fraction R (33%) onto a CCDcamera (Retiga EXi, QImaging). The remaining signal wasfocused through a 25 µm pinhole onto an avalanche photo-diode (APD) (SPCM-AQR-14, Perkin Elmer). The trackingalgorithm was implemented on a CompactRIO (cRIO-9076,National Instruments), consisting of a field programmablegate array (FPGA), a real-time processor, a 16-bit 100 kHzanalog input module (NI 9215), a 16-bit 100 kHz analogoutput module (NI 9263), and a digital input/output module(NI 9401). The analog input module measured stage positionfeedback and the analog output module commanded an actu-ation voltage to the stage amplifier. The digital input/output

51st IEEE Conference on Decision and ControlDecember 10-13, 2012. Maui, Hawaii, USA

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module was connected to the digital output of the APD.

Fig. 1. Diagram of confocal microscope used in the tracking experiments.The excitation laser (thick blue line) was focused by an objective lensinto the sample. Fluorescence (thin green line) generated by beam/sampleinteractions was passed through the objective and the dichroic. A beamsplitter reflected a fraction R of the fluorescent light to a CCD cameraand passed the remaining light to the pinhole. The light focused onto thepinhole was detected by an avalanche photodiode. The sample was displacedby a three-axis piezoelectric nanostage and the tracking algorithm wasimplemented on a National Instruments CompactRIO.

III. TRACKING ALGORITHMRecall that a confocal microscope uses a diffraction-

limited excitation source to illuminate points of afluorescently-labeled specimen. When imaging a particlemuch smaller than the diffraction limit (i.e. a point source),the spatial intensity pattern formed at the detector is the pointspread function (PSF) of the microscope. Although an analyt-ical expression for the PSF can be derived [11], it is difficultto incorporate into real-time control algorithms due to itscomplexity. Consequently, our algorithm treats the PSF asan unknown scalar potential field and therefore assumes verylittle knowledge of the underlying structure. An importantfeature of an aberration-free PSF utilized by the algorithm isthe existence of a maximum when the focus of the laser isperfectly centered on the particle. The algorithm, (2) below,adds artificial non-holonomic constraints to the motion of thenanopositioning stage and drives the trajectories to a volumearound the maximum provided a sufficiently large signal-to-noise ratio (SNR). As shown in Fig. 2, the algorithm consistsof two components: a slow outer loop for position commandgeneration and a fast inner loop for stage position actuation.

The outer loop generates desired position commands basedon the change in the measured intensity along the trajectory.It is described briefly below; details can be found in [8]. Letthe desired velocities of the (x,y,z) stages respectively be

u = (ux uy uz)T . (1)

Given the intensity I(t), the discrete-time control law is givenby

θ(t) = θ(t−∆t)+ vr(∆t−K∆I(t)) , (2a)

φ(t) = φ(t−∆t)+ωz∆t, (2b)∆I(t) = I(t)− I(t−∆t), (2c) ux(t)uy(t)

uz(t)

= v cosθ(t)sinθ(t)cosφ(t)−sinθ(t)sinφ(t)

(2d)The control law has several tunable parameters, including

the control update period ∆t, the proportional gain K, they− z plane rotation rate ωz, the speed v, and the nominalradius r. All parameters are assumed to be strictly positive.

Although not apparent in (2), the intensity accumulationtime significantly affects the likelihood of convergence in thepresence of measurement noise. For example, short accumu-lation times reduce the SNR and better approximate a point-measurement, whereas longer accumulation times increaseSNR but represent an average over a sweep of positions. Thechoice of accumulation time reflects the unavoidable tradeoffbetween speed and sensitivity and is common to all particletracking algorithms.

Lastly, the stages must follow the commanded trajectorieswith sufficiently small error, so, assuming sufficient SNR inthe intensity measurements, the minimum bandwidth of theinner loop ultimately determines the quality of tracking.

ActuatorController

ActuatorPlant

ActuatorPositionSensing

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Fig. 2. High-level block diagram of the tracking algorithm. The trackingcontroller generates position commands based on changes in intensity. Thefast inner loop responds to position commands generated by the trackingcontroller.

The outer loop assumes the positioning stages follow thecommanded signals faithfully. Thus, tracking performanceis limited by the closed-loop bandwidths of the stages.To reduce tracking error at higher frequencies, a customfeedback controller was designed and implemented in theFPGA on the CompactRIO.

System identification of each stage was performed bymeasuring the plant frequency responses through randomexcitation. A pseudorandom binary sequence with a 0.01 Vamplitude was driven into each stage and position feedbackwas measured at a rate of 100 kHz. The position feedbacksignal was assumed to be corrupted by noise, so the H1estimator was calculated using Bartlett’s method [12].

Given these frequency responses, a controller was de-signed for each stage using loop shaping principles. Toreduce quantization noise, which is especially prevalent infixed-point systems, a low-order controller structure was

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chosen and consisted of a gain, a pure integrator, andtwo first-order low-pass filters. Further, dynamic couplingwas assumed to be negligible given the “frame-in-frame”design of the stages. The corner frequencies of the low-passfilters were chosen to robustly attenuate the high-frequencyresonances and the gains were selected to set the open loopcrossover frequencies. Each feedback controller was shapedin continuous-time and discretized for a sampling frequencyof 100 kHz using the bilinear transform. The resultingbandwidths of the complementary sensitivities were (800,200, 100) Hz in (x,y,z), respectively. Frequency responsesdetailing stability and performance are shown in Fig. 3.

IV. LOCALIZATION OF A DIFFUSING PARTICLE

Since the control law is reactive in nature it does not trackby localizing position. While this leads to a simple algorithmthat is tunable to perform well at different intensities, SNRs,and rates of dynamics, it does mean that position must beinferred from the data. One approach of analyzing trackingperformance is to estimate the particle trajectories usingoffline localization. Since the measurements are intensitiesalong a 3-D trajectory, rather than images, fitting techniquescommon to wide-field approaches such as Gaussian fitting[13] or maximum likelihood [14] are not applicable. We turninstead to the fluoroBancroft (FB) algorithm developed byone of the authors [10]. This approach provides an analyticalsolution to the estimation problem using as few as four nonco-planar measurements. The FB algorithm approximatesthe PSF as a Gaussian function with parameters σx, σy, σz,corresponding to the spread widths. From the results derivedin [15], we assume

σx = σy =√

2λ2π (N.A.)

, σz =2√

6nλ2π (N.A.)2

, (3)

where λ is the fluorescence wavelength, N.A. is the numer-ical aperture of the lens, and n is the refractive index of theimmersion fluid.

The position of a diffusing particle xp ∈ R3 sampled attime step δ t can be modeled by the first-order stochasticdifference equation

xp[k+1] = xp[k]+√

2D(δ t)w[k], (4)

where D is the diffusion coefficient and w[·] is a Gaussianwhite noise process with unit variance and zero mean. Toestimate the position of the particle in time, we assume theparticle is moving slowly relative to the motion of the stagesand apply the FB algorithm to a finite set of sequentialmeasurements. In other words, for a measurement vectorconsisting of intensity and corresponding stage positionsγ[k] = (I[k] x[k] y[k] z[k])T , we estimate the particleposition x̂p[k] by applying the FB algorithm to the set{γ[k],γ[k+1], . . . ,γ[k+N−1]}, where N is the FB windowlength. Elements with intensities less than or equal to themean of the background noise are discarded. When used inthis manner, the FB algorithm behaves like a low-pass filterwith N inversely proportional to the cut-off frequency. Thus,N should be maximized to yield a large SNR yet chosen

small enough so that the particle does not move significantlywithin the time period. The estimated particle positions aresmoothed using a Rauch-Tung-Striebel (RTS) smoother [16]and the mean square displacement (MSD) is calculated [17].The FB window length and the RTS process variance Q areadjusted until the MSD of the estimated particle position isapproximately linear. Recall that for a particle undergoingpure diffusion the MSD is linear with respect to time. Thediffusion coefficient is then estimated by fitting a line to aportion of the MSD.

V. EXPERIMENTAL RESULTS

In this section, we present experimental results of thetracking algorithm when applied to diffusing quantum dots(QD625, Life Technologies) in solutions of differing vis-cosities. Samples were prepared by diluting quantum dotsin deionized water and by adding different proportions ofglycerol.

A. Tracking a Fixed Particle

To obtain a fixed quantum dot, a 9:1 solution of glycerolto water was created and dried onto a cover slip. The coverslip was adhered to a microscope slide and mounted ontothe stage. A fixed quantum dot was found using the CCDand the tracking algorithm was enabled. The control updaterate was set to 2 kHz and the APD accumulation timewas set to 500 µs. The velocity v was 10 µm/s and thesteady-state radius r was 20 nm, corresponding to 75 rev/saround the steady-state circle. The radius was chosen smallrelative to the width of the PSF (calculated below) to yieldhigh sensitivity and the velocity was chosen to yield arotational frequency large enough so the circle was traversedin minimal time yet small enough that the stages couldfollow the trajectory. The speed of the rotating plane ωz was30 rev/s and the gain K was 0.1, both of which were adjustedexperimentally. The laser power was approximately 4 mW.

Fig. 4 shows the x and z traces for a typical run. (Thetrace in the y–axis was nearly identical to that in x and wasomitted for space reasons.) The intensity was regulated to38±6 counts/500 µs, representing the mean and standarddeviations. The standard deviation matches what one wouldexpect for shot noise. The position of the quantum dot wasestimated by the procedure described in Sec. IV. Since thequantum dots fluoresce with a wavelength of 625 nm, theFB parameters were chosen to be σx = σy = 117 nm andσz = 451 nm. The mean background noise ηB was measuredto be 0.1 counts/500 µs when no quantum dot was present.The FB window length was 500 points (corresponding to250 ms), the RTS process noise variance Q was 0.01 µm2,and the measurement noise variance R was unity. These pa-rameters were adjusted until the trajectories were adequatelysmoothed.

Correlation between the (x,y) trajectories was observed,indicating the presence of a non-random disturbance such asmechanical or sensor drift due to a lack of thermal regulationor fluidic flow caused by laser-induced heating. Moreover,the z trajectory appeared significantly different from the x

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Fig. 3. Frequency responses for the (x,y,z) stages. Discrete-time linear feedback controllers were designed for each stage using loop-shaping principlesand were implemented in an FPGA at a 100 kHz update rate. The bandwidths of the complementary sensitivity functions (shown in red) were approximately(800,200,100) Hz in (x,y,z), respectively.

trajectory due to the small commanded radius which impliesthat most of the measurements were observed in the (x,y)plane. As such, there was larger uncertainty in z.

B. Tracking Diffusing Particles

To determine how well the controller could track a slowlydiffusing particle, a small amount of the previously men-tioned 9:1 solution was placed into a multiwell microscopeslide, sealed with a cover slip, and mounted to the stage.A diffusing dot was located with the CCD and the trackingalgorithm was turned on. The control update rate was set to5 kHz and the APD accumulation time was set to 200 µs.The velocity v was 750 µm/s and the steady-state radius rwas 637 nm, corresponding to 187.5 rev/s around the steady-state circle. The radius was chosen to enlarge the searchspace of the controller and the velocity was chosen to setthe nominal rotational frequency. The chosen rotational fre-quency surpassed the bandwidth of the z stage, but trackingwas maintained at the cost of reducing the effective radius inz. The speed of the rotating plane ωz was 137.5 rev/s and thegain K was 0.4, both of which were chosen experimentally.The laser power was approximately 4 mW.

The results are shown in Fig. 5. The stages tracked theparticle for more than 20 s with an average intensity of2 counts/200 µs. The experiment had ended before trackingwas lost. Although the average intensity was low, the max-imum observed intensity was over 80 counts/200 µs. Thisdiscrepancy is due to the large radius and short accumulationtime. The position of the quantum dot was estimated asdescribed in the previous subsection. The mean backgroundnoise was measured to be approximately zero and was ne-glected. The window length was 150 points (corresponding to30 ms), the RTS process noise variance was 2·10−6 µm2, andthe measurement noise variance was unity. These parameterswere adjusted to yield linearity in the MSD. A linear fitto the MSD yielded a diffusion coefficient of 0.24 µm2/s.Since the commanded radius was chosen much larger thanthe assumed PSF widths, estimation errors occurred because

the PSF is not truly Gaussian at large distances from thecenter of the particle. In addition, a relatively low SNR and asuboptimal measurement constellation [18] resulted in noisyposition estimates.

To stress the tracking algorithm and explore the limit onfaster particles, a new 5:1 solution of glycerol to water wasmade and prepared as described in the previous section.The control update rate was set to 10 kHz and the APDaccumulation time was set to 100 µs. The velocity v was750 µm/s and the steady-state radius r was 1.193 µm,corresponding to 100 rev/s around the steady-state circle.The speed of the rotating plane ωz was 45 rev/s and thegain K was 10, both of which were selected experimentally.To improve the SNR, the laser power was increased toapproximately 20 mW. The resulting trajectories are shownin Fig. 6. The stages tracked the quantum dot for morethan 5 s with an average intensity of 0.3 counts/100 µs.Again, the experiment ended before tracking failed. Theposition of the quantum dot was estimated as previouslydescribed. The mean background noise was measured to beapproximately zero and was neglected. The window lengthwas 150 points (corresponding to 15 ms), the RTS processnoise variance was 10−5 µm2 and the measurement variancewas 1. A line fit to the MSD yielded a diffusion coefficientof 3.7 µm2/s. The FB estimates were noisier due to theincreased radius and the decreased accumulation time. Assuch, the uncertainty in the position estimates, and thereforethe diffusion coefficient, was significant.

VI. DISCUSSION

Given the results detailed in Sec. V, it is evident thatthe control law tracks diffusing quantum dots with verylittle information about the PSF. Moreover, the sensitivityand speed of the controller can be adjusted using intuitiveparameters. For example, the controller used to track thefixed quantum dot was tuned for high-sensitivity with asmall radius of 20 nm and a long accumulation time of500 µs. This allowed the controller to stay very close to

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Fig. 4. Tracking run with a fixed particle. Plots (a)-(b) show (x,z) stage trajectories (blue), fluoroBancroft estimates of the particle (green), and smoothedfluoroBancroft estimates (red). The y trajectory was nearly identical to x and was therefore omitted. Plot (c) shows the measured intensities in counts/500 µs.The controller regulated the intensity to 38±6 counts/500 µs with the mean indicated by the solid red line in (c).

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Fig. 5. Tracking run with a slowly diffusing particle. Plots (a)-(c) show stage trajectories (blue), fluoroBancroft estimates of the particle (green), andsmoothed fluoroBancroft estimates (red center line). Plot (f) shows a 3-D image of the stage trajectories (green) and the smoothed positions (blue, withprojections onto respective planes). Plot (e) shows the measured intensities with an average (indicated by the red line) of 2 counts/200 µs. The mean squaredisplacement is shown in (d) as a red curve with corresponding linear fit (blue) indicating a diffusion coefficient of 0.24 µm2/s.

the maximum of the PSF. The other controllers were tunedfor high-speed with larger radii and shorter accumulationtimes. Although we showed the controller tracking a quantumdot diffusing at 3.7 µm2/s, it is unknown whether this isan upper limit. A rigorous understanding of stability andperformance could yield optimal parameters for improvedtracking. Further, since the outer loop assumes the innerloop faithfully follows the command signals, more accuratetracking could be achieved by raising the bandwidth ofthe inner loop. This could be done by implementing moreadvanced feedback or feedforward controllers or by usingfaster actuators.

Although the controller avoids temporal inefficiencies bynot directly estimating the position of the particle, calculationof diffusion coefficients must be performed offline using

stage trajectories and intensities. Here we used the FBalgorithm to estimate the particle positions. The choice ofwindow length in the estimation is essential as choosingit too short leads to noisy estimates while choosing it toolong leads to a low-pass filtering effect and thus an under-reporting of the diffusion coefficient. It is possible thatbetter estimation can be achieved by optimizing the scanparameters with such estimation in mind, possibly whilesacrificing some tracking speed. A different approach is totake advantage of the residual fluctuations in the measuredintensity as these encode information about the correlationstatistics of the particle. This approach has been developedsuccessfully under a different tracking scheme and allowsfor a direct computation of the diffusion coefficient withoutever estimating the actual particle position [19].

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Fig. 6. Tracking run with a quickly diffusing particle. Plots (a)-(c) show stage trajectories (blue), fluoroBancroft estimates of the particle (green - noisylines), and a smoothed fluoroBancroft estimate (red center line). Plot (f) shows a 3-D image of the stage trajectories (green) and the smoothed estimates(blue, with projections onto respective planes). Plot (e) shows the measured intensities with an average (indicated by the red line) of 0.3 counts / 100 µs.The mean square displacement is shown in (d) as a red curve with corresponding linear fit (blue) indicating a diffusion coefficient of 3.7 µm2/s.

VII. CONCLUSIONIn this paper we demonstrate the effectiveness of a non-

linear reactive control law that tracks particles in threedimensions. Experimental results indicate the controller iscapable of tracking diffusing quantum dots on a standardconfocal microscope without directly estimating the positionof the particle and without any a priori knowledge of theoptical setup. Since the particle position is not used by thecontroller, performance validation and diffusion statistics canbe calculated offline by analyzing the resulting trajectoriesand intensities. Moreover, the controller can be tuned fordifferent sensitivities and speeds, thereby increasing its ef-fectiveness of tracking particles in various conditions.

VIII. ACKNOWLEDGEMENTSThis work was supported in part by NSF through grant

CMMI-0845742.

REFERENCES[1] S.-L. Liu, Z.-L. Zhang, Z.-Q. Tian, H.-S. Zhao, H. Liu, E.-Z. Sun,

G. F. Xiao, W. Zhang, H.-Z. Wang, and D.-W. Pang, “Effectively andefficiently dissecting the infection of influenza virus by quantum-dot-based single-particle tracking,” ACS Nano, vol. 6, no. 1, pp. 141–150,2012.

[2] X. Hao, X. Shang, J. Wu, Y. Shan, M. Cai, J. Jiang, Z. Huang, Z. Tang,and H. Wang, “Single-particle tracking of hepatitis B virus-like vesicleentry into cells,” Small, vol. 7, no. 9, pp. 1212–1218, 2011.

[3] E. J. G. Peterman, H. Sosa, and W. E. Moerner, “Single-moleculefluorescence spectroscopy and microscopy of biomolecular motors,”Annu. Rev. Phys. Chem, vol. 55, pp. 79–96, 2004.

[4] K. McHale, A. J. Berglund, and H. Mabuchi, “Quantum dot photonstatistics measured by three-dimensional particle tracking,” Nano Lett.,vol. 7, no. 11, pp. 3535–3539, 2007.

[5] Z. Shen and S. B. Andersson, “3-D tracking of fluorescent nanoparti-cles in a confocal microscope,” in Proc. of the IEEE Conf. on Decisionand Control, 2011, pp. 5856–5861.

[6] H. Cang, C. S. Xu, D. Montiel, and H. Yang, “Guiding a confocalmicroscope by single fluorescent nanoparticles,” Opt. Lett., vol. 32,no. 18, pp. 2729–2732, 2007.

[7] N. P. Wells, G. A. Lessard, and J. H. Werner, “Confocal, three-dimensional tracking of individual quantum dots in high-backgroundenvironments,” Anal. Chem., vol. 80, no. 24, pp. 9830–9834, 2008.

[8] S. B. Andersson, “A nonlinear controller for three-dimensional track-ing of a fluorescent particle in a confocal microscope,” Appl. Phys. B,vol. 104, no. 1, pp. 161–173, 2011.

[9] D. Baronov and S. B. Andersson, “Controlling a magnetic force micro-scope to track a magnetized nanosize particle,” IEEE T. Nanotechnol.,vol. 9, no. 3, pp. 367–374, 2010.

[10] S. B. Andersson, “Localization of a fluorescent source without numer-ical fitting,” Opt. Expr., vol. 16, no. 23, pp. 18 714–18 724, 2008.

[11] J. E. Jonkman and E. H. K. Stelzer, Resolution and Contrast inConfocal and Two-Photon Microscopy. Wiley-Liss, 2002, pp. 101–125.

[12] J. S. Bendat and A. G. Piersol, Random Data: Analysis and Measure-ment Procedures, 3rd ed. Wiley, 2000, pp. 189–217.

[13] R. E. Thompson, D. R. Larson, and W. W. Webb, “Precise nanometerlocalization analysis for individual fluorescent probes,” Biophys. J.,vol. 82, no. 5, pp. 2775–2783, 2002.

[14] K. I. Mortensen, L. S. Churchman, J. A. Spudich, and H. Flyvbjerg,“Optimized localization analysis for single-molecule tracking andsuperresolution microscopy.”

[15] B. Zhang, J. Zerubia, , and J.-C. Olivo-Marin, “Gaussian approxima-tions of fluorescence microscope point-spread functions,” Appl. Opt.,vol. 46, no. 10, pp. 1819–1829, 2007.

[16] D. Simon, Optimal State Estimation: Kalman, H Infinity, and Nonlin-ear Approaches. Hoboken: John Wiley and Sons, 2006, pp. 286–294.

[17] M. J. Saxton and K. Jacobson, “Single-particle tracking: applicationsto membrane dynamics,” Ann. Rev. Biophys. Biomol. Struct., vol. 26,pp. 373–399, 1997.

[18] Z. Shen and S. B. Andersson, “Optimal measurement constellationof the fluorobancroft localization algorithm for position estimation intracking confocal microscopy,” Mechatronics, 2011. In press.

[19] A. J. Berglund and H. Mabuchi, “Tracking-FCS: fluorescence correla-tion spectroscopy of individual particles,” Opt. Expr., vol. 13, no. 20,2005.

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